Problem: The number of mosquitoes in Brooklyn (in millions of mosquitoes) as a function of rainfall (in centimeters) is modeled by $m(x)=-x(x-4)$ What is the maximum possible number of mosquitoes?
Answer: The number of mosquitoes is modeled by a quadratic function, whose graph is a parabola. The maximum number is reached at the vertex. So in order to find the maximum number, we need to find the vertex's $y$ -coordinate. We will start by finding the vertex's $x$ -coordinate, and then plug that into $m(x)$. The vertex's $x$ -coordinate is the average of the two zeros, so let's find those first. $\begin{aligned} m(x)&=0 \\\\ -x(x-4)&=0 \\\\ \swarrow &\searrow \\\\ -x=0\text{ or }&x-4=0 \\\\ x={0}\text{ or }&x={4} \end{aligned}$ Now let's take the zeros' average: $\dfrac{({0})+({4})}{2}=\dfrac42= 2$ The vertex's $x$ -coordinate is $ 2$. Now let's find $m({2})$ : $\begin{aligned} m( 2)&=-( 2)( 2-4) \\\\ &=-(2)(-2) \\\\ &=4 \end{aligned}$ In conclusion, the maximum possible number of mosquitoes is $4$ million.